Slope is defined by a plane tangent to a topographic surface, as modelled
by the DEM at a point (Burrough, 1986). Slope is classified
as a vector; as such it has a quantity (gradient)
and a direction (aspect). Slope gradient is defined as the maximum rate of
change in altitude (tan Q), aspect
(y) as the compass direction of this maximum rate of change (Cf. Fig.
1). More analytically, slope gradient at a point is
the first derivative of elevation (Z) with respect of the slope (S), where
S is in the aspect direction (y). At the
same time the first derivative of a function (i.e. S stands for slope) at
a point can be defined as the slope (angular coefficient or trigonometric
tangent) of the tangent to the function on that particular point, hence:

2.Different techniques for calculating gradient and aspect

The technique/algorithm adopted to calculate
slope gradient and aspect varies according to the type of DEM selected to
model the topography. Models for structuring elevation database can be square-grid,
TIN (triangular irregular network), and contour-based ones (Check here
for more details on DEMs).

After conducting a literature review
(though not a comprehensive one), it appears that a greater number of
algorithms have been developed for gridded DEMs (Cf. Table 1).

A study conducted by Srinivasan and Engel
(1991) compared methods 3, 4, 5 and 6 (Cf. Table
1) against field measurements. According to their results the neighbourhood and quadratic surface methods would follow more faithfully ground
measurements, the maximum slope
method being the least reliable.

N.B. DEM derivatives are also highly dependent on the type of
interpolation adopted during the process of DEM creation (e.g. kriging, spline,
etc.).

3.Applications

Among a number of possible applications,
I’d like to stress in a Pacific Northwest context, how important is to
automatically derive slope gradient and aspect in the prospective to compile
more sophisticated thematic maps such as: